Asalaamu Alaikum(peace be with you),
We don't differentiate between our Prophet's(pbut), so if we did celebrate the birth of one Prophet, we'd have to do it for all Prophets(pbut). But more importantly, our final messenger(pbuh) stressed that like all humans, he(pbuh) was but a Slave of God and nothing more. Here's some references;
The Prophet (sallallahu 'alaihi wa sallam) has said: Do not over praise me as the Christians over-praised the son of Mary. I am His slave so say: 'Allah's slave and messenger' [Al-Bukhaari and Muslim]
And those who believe in Allâh and His Messengers and make no distinction between any of them (Messengers), We shall give them their rewards, and Allâh is Ever Oft*Forgiving, Most Merciful. [4:152]
A desert Arab came to Allah's Apostle
and asked him about ablution. He demonstrated (washing each part of his body) thrice, and then said: That is (the method) of the ablution. And he who does more than this has done wrong, transgressed the limit and has oppressed (himself). [Transmitted by Nasa'i, Ibn Majah, and Abu Dawud transmitted something to the same effect.]
O people of the Scripture (Jews and Christians)! Do not exceed the limits in your religion, nor say of Allâh aught but the truth. ...[4:17]
The Prophet
came to Medina with two days they played in.
The Prophet
said, ‘What are these two days?’ They said, ‘These are two days we used to play in, in our Jahiliyah.’ The Prophet
said, ‘Allah has replaced them with two better days: Eid Al Adhaa and Eid Al-Fitr’. [
Abu Dawood #1134, 1/675, Narrated Anas Bin Malik]
In Conclusion, no - Muslims don't celebrate Christmas.